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Periodic Solution and Ergodic Stationary Distribution of Stochastic SIRI Epidemic Systems with Nonlinear Perturbations

  • Weiwei Zhang
  • Xinzhu MengEmail author
  • Yulin Dong
Article
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Abstract

This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations. The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds. For the nonautonomous stochastic SIRI epidemic system with white noise, the authors provide analytic results regarding the stochastic boundedness, stochastic permanence and persistence in mean. Moreover, the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory. For the system with Markov conversion, the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution. In addition, sufficient conditions for the extinction of disease are obtained. Finally, numerical simulations are introduced to illustrate the main results.

Keywords

Extinction and stochastic permanence Markov chain periodic solution stationary distribution and ergodicity stochastic SIRI epidemic model 

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Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Electrical Engineering and AutomationShandong University of Science and TechnologyQingdaoChina
  2. 2.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  3. 3.State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and TechnologyShandong University of Science and TechnologyQingdaoChina

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